MECHANICS line. Let OA, AB (fig. 20) be one of these. Then, in any other position, OP and PQ are equally inclined to OA. The path of Q is an ellipse, of which the major semi-axis OB is the sum of the radii, and the minor axis their difference. Hence when the radii are equal the result is simple harmonic motion in the line OBB Thus we have the proposition, of very great importance in optics, that a simple harmonic motion may B/ be looked upon as the resultant of two equal and opposite circular motions in one plane. r %- 20 - When the periods are not exactly equal, the motion may be regarded as simple harmonic motion, in a line which rotates with uniform angular velocity in a plane. This is the case of Foucault s pendulum, and of plane polarized light passing along the axis of a crystal of quartz, or through a piece of glass or other transparent substance in the magnetic field. cloids. g 66. Uniform circular motions, of different periods, give epicycloids, &c. A particular case is uniform circular motion superposed on uniform rectilinear motion, in which case we have cycloids, &c. But these we merely mention. 67.. By far the most important of the applications of simple harmonic analysis is summed up in what is called I trier s FOURIER S THEOREM. A complex harmonic function, orem- luith a constant term added, is the proper expression for any periodic single-valued function, and, consequently, can express any single-valued function whatever between any assigned values of the variable. To show the importance of this in physics we need take but a single example. The one essential characteristic of a musical sound is its " periodicity." Hence it may be analysed into a series of simple harmonic disturbances. Their respective periods are the fundamental period, its half, third, fourth part, &c. The first gives the pitch of the note ; the others determine its quality. The investiga tion which follows is not intended to prove the theorem ; it is merely introduced as readily suggesting it. The essence of periodicity of a function /is that we must have f[x + d =f[x - |a] whatever be x, provided a be the period. We may write this as ad ad or / ^L ^ _ ^ ^_
2 dx - g idx. )f(x] = .
Now the equation has the real root and the infinite series of pairs of imaginary roots where i is any integer. Hence 6 V) . . . so that the differential equation for f(x) gives, besides a constant term, the infinite series of terms due to solutions of equations of the second order of which the type is d The solution of this representative equation gives the following particular integral of the complete equation, f(x) = P,- cos (2ivxa - l + Q,-) . Hence the general solution is f(x] = A + 2* Pi cos (Ziirxa - 1 + Q,-) = A + 2* 689 where the constants are to be determined by special integration, according to the process already described in the article HAUMONIO ANALYSIS (q.v.). As a single example, suppose that the value of f(x] is unity from x = Q to x = a, and zero from x = a to x = 2a. This has many applications, as, for instance, to alternate heating and cooling of one surface of a solid, alternate " make and break " with a battery and a telegraph wire, &c. In this case we have J(x) -jr-- 2, . . sin - 7T 68. A point describes a logarithmic spiral with constant Resisted angular velocity about the pole ; find the acceleration. 1 harmonic Since the angular velocity of SP (fig. 21) and the incli- motion - nation of this line to the tangent are each constant, the linear velocity of P is as SP. Take a length PT, equal to e.SP, to represent it. Then the hodograph, the locus of p, where S/> is parallel and equal to PT, is evidently another logarithmic spiral, similar to the former, and described with the same con stant angular velocity. Hence pt, the acceleration required, is equal to e.Sp, and makes P > with Sp an angle equal to SPT. Hence, if Pu be drawn parallel and equal to pt, and uv parallel to PT, the whole Fi S- 21. acceleration Pu may be resolved into Pv and vu ; and Pvu is an isosceles triangle, whose base angles are each equal to the angle of the spiral. Hence Pv and vu bear constant ratios to Pu, and therefore also to SP or PT. The acceleration, therefore, is composed of a central acceleration proportional to the distance, and a tangential retardation proportional to the velocity. And, if the resolved part of P s motion parallel to any line in the plane of the spiral be considered, it is obvious that in it also the acceleration will consist of two parts one directed towards a point in the line (the projection of the pole of the spiral) and proportional to the distance from it, the other proportional to the velocity but retarding the motion. Hence a particle which, unresisted, would have a simple harmonic motion has when subject to resistance propor tional to its velocity a motion represented by the resolved part of the spiral motion just described. If a be the angle of the spiral, to the angular velocity of SP, we have evidently PT. sina = SP.w. Hence SP bin a and 3P = ?i 2 . SP (suppose); . PT (suppose). Thus the central acceleration at unit distance is 2 = w 2 /sin 2 a, and the coefficient of resistance is 2& = 2wcosa/sina. The time of oscillation is evidently 27r/(o ; but, if there had been no resistance, the properties of simple harmonic motion show that it would have been 27r/. ; so that it is increased by the resistance in the ratio coseca : 1, or n : *Jn 2 & 2 . The rate of diminution of SP is evidently nm wCOSa 7Cir , PT . coso= : - SP = &SP ; that is, SP diminishes in geometrical progression as time increases, the rate being k per unit of time per unit of length. By an ordinary result of arithmetic (compound interest payable every instant) the diminution of log SP in unit of time is L 1 The physical application of this problem to pendulum motion, taking place in a medium in which there is resistance proportional to the velocity, will be afterwards discussed analytically.
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